2dof pid controller design for a class of foptd models an analysis with heuristic algorithms

Peer-review under responsibility of scientific committee of International Conference on Computer, Communication and Convergence ICCC 2015 doi: 10.1016/j.procs.2015.04.155 ScienceDirect

Procedia Computer Science 48 ( 2015 ) 90 – 95

1877-0509 © 2015 The Authors Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license

( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

Peer-review under responsibility of scientific committee of International Conference on Computer, Communication and Convergence (ICCC 2015) doi: 10.1016/j.procs.2015.04.155

ScienceDirect

International Conference on Intelligent Computing, Communication & Convergence

(ICCC-2014) Conference Organized by Interscience Institute of Management and Technology,

Bhubaneswar, Odisha, India

2DOF PID Controller Design for a Class of FOPTD Models – An

Analysis with Heuristic Algorithms

K Sundaravadivua,* , S Sivakumara, N Hariprasada

a St Joseph’s College of Engineering, Department of Electronics and Instrumentation Engineering, Chennai 600 119, India

Abstract

In recent years, a number of controller design procedures are developed and implemented in process industries to enhance the performance of closed loop processes In this paper, heuristic algorithm based Two Degrees Of Freedom (2DOF) PID controller design is proposed for a class of First Order Plus Time Delay (FOPTD) systems existing in the literature Minimization of the weighted sum of multiple objective functions is considered to monitor the heuristic search towards the optimal controller parameters A detailed comparative analysis between well known heuristic methods, such as Particle Swarm Optimization (PSO), Bacterial Foraging Optimization (BFO), Cuckoo Search (CS) and Firefly Algorithm (FA) are presented The popular 2DOF PID structures, such as Feed Back Structure (FBS) and Feed Forward Structure (FFS) are considered in this work to enhance the performance of FOPTD systems From the results, it is noted that, proposed controller provides enhanced results for the reference tracking and disturbance rejection operations

© 2014 The Authors Published by Elsevier B.V

Selection and peer-review under responsibility of scientific committee of Missouri University of Science and Technology

Keywords: FOPTD; 2DOF PID controller; heuristic algorithm; reference tracking; disturbance rejection

1 Introduction

* Corresponding author Tel.: +91 9884691413

E-mail address: ksvadivud@gmail.com

© 2015 The Authors Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license

( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

Peer-review under responsibility of scientific committee of International Conference on Computer, Communication

and Convergence (ICCC 2015)

(ICCC-2015)

In process industries, despite the major progress in superior process control methodologies, PID controllers are widely used because of its structural simplicity, reputation, easy in performance, acceptance simplicity and adaptability [1, 6] In the literature, several articles are available to study the tuning procedures and implementation

of a single Degree Of Freedom (1DOF) PID controller for stable, unstable and nonlinear systems [1,7,9] The recent studies on fine tuning the 1DOF PID have provided insight for better understanding of the controller performance for a class of process models For most of the systems, 1DOF PID offers a feasible outcome either for reference tracking operation or disturbance rejection operation

Nomenclatures

α, β Tuning parameters

J min Objective function to be minimized

Abbreviations

PID Proportional + Integral + Derivative

In recent years, various forms of Two Degrees Of Freedom (2DOF) PID controllers are widely discussed by the researchers [6,9,] A detailed study on various 2DOF structures existing in the literature can be found in the article

by Araki and Taguchi [2].

Most of the conventional controller tuning methods existing in the literature is purely model dependent The tuning methodology employed for one particular reduced process model may not offer a suitable response for other process models Hence, in recent years, heuristic algorithm based model free controller design procedure is widely adopted

by the researchers [6,8-10]

In the proposed work, popular 2DOF PID structures, such as Feed Back Structure (FBS) and Feed Forward Structure (FFS) are considered to stabilize First Order Plus Time Delay (FOPTD) models existing in the literature using heuristic algorithms, such as PSO, BFO, CS and FA The performances of the considered algorithms are

analyzed based on the time domain parameters (M p , t s ), error values (ITSE, ITAE) and the search time taken by the

algorithms

In general, 2DOF PID structure improves the overall closed loop performance of the process A detailed study on various 2DOF structures are clearly presented by Araki and Taguchi [2] In this work, the 2DOF PID structures considered by Latha and Rajinikanth [6] is adopted to stabilize the FOPTD process models

) ( D K ) 1 ( K ) 1

(

K

)

(

C 2 (s )

R(s)

_ +

D(s)

+ +

Fig 1 Feedback structure

FOPTD Y(s) R(s)

C 3 (s )

E (s)

U(s) _ +

D(s)

+ +

C 4 (s )

Fig 2 Feed forward structure

D ( ) K K D ( )

K

)

(

) s ( D K K K ) s ( D s

1 1 K

)

s

(

i p

¸¸¹

·

¨¨©

§



D ( ) K K D ( )

K

)

(

where α and β are controller weighting parameters ranging from 0 to 1 and Df( ) is the derivative filter term given by s /( 1  Nfs ) In this work, N f is chosen as 20

Fig 1 depicts the feedback 2DOF structure with a PD controller in the inner loop and a PID in the outer loop In

this structure, the PID responds on error signal e(t) and the PD works on the process output y(t) Fig 2 shows the

feed forward 2DOF structure with a PD in the feed forward loop and a in the closed loop The PID controller

responds on error signal e(t) and the PD controller works on the reference input r(t) The major advantage of the

2DOF structure is, it is free from the proportional and derivative kick effect and supports smooth reference tracking response

3 Heuristic Algorithms in this Study

In recent years, a considerable number of heuristic algorithms are proposed by the researchers to find optimal solutions for a class of engineering optimization problems The details of the existing heuristic algorithms can be found in the recent article by Fister et al [4] In this paper, the following heuristic algorithms are considered to offer optimal 2DOF PID controller parameters

3.1 Particle swarm optimization

PSO is an evolutionary optimization technique, developed due to the inspiration of the social activities in flock of birds and school of fish [5] It has two basic equations namely the velocity update and position update as given below:

) S G ( R C ) S P ( R C V W

) 1 t i t i )

1

t

where Wt= inertia weight coefficient (typically 0.75), ti V = current velocity of particle, V i ( t1 )= updated

velocity of particle, C 1 = 2.1, and C 2 = 1.8 [9]

3.2 Bacterial foraging optimization

BFO is developed by mimicking the foraging behavior of E coli bacteria [8] In this work, the enhanced BFO

algorithm discussed by Rajinikanth and Latha is adopted [9,10] The initial algorithm parameters are assigned as follows:

Number of E.Coli bacteria = N

N c =

2

N

; N s = N re |

3

N

; N ed |

4

N

; N r =

2

N

Ped = ¨¨©§N  r N ¸¸¹·

ed N

;

dattractant = Wattractant =

N

; and hrepellant = Wrepellent =

N

3.3 Cuckoo search

CS was initially proposed by Yang and Deb in 2009 [14] This algorithm is based on the breeding tricks of parasitic cuckoos In CS, the new solution (X i t1 )) mainly depends on the old solution (X i ( )) and the search guiding procedure In this work, Lévy Flight (LF) based search is considered as presented below:

LF X

3.4 Firefly algorithm

FA is a nature inspired metaheuristic algorithm proposed by Yang [15,16] This algorithm is developed by imitating the flashing illumination patterns produced by invertebrates such as glowworm and firefly [17] A detailed analysis

on FA can be found in [17] In this work, LF based firefly discussed is literature is adopted and the following update equation is considered:

Lévy

½) sign(rand

) X X ( e β X

2 ij d γ 0 t i 1

t

3.5 Implementation

Fig 3 shows the proposed controller tuning procedure The heuristic algorithm is employed to find the best possible

values of 2 DOF PID parameters , such as K p , K i , K d , α, and β In this work, the dimension of the search is chosen

as five

In heuristic algorithm based search procedure, the optimization accuracy mainly relies on the cost function assigned

to guide the search In this paper, a weighted sum of cost function shown below is considered:

ITAE

* w ITSE

* w t

* w M

*

w

Jmin 1 p 2 s 3  4

where the weights are chosen as w1 =w2 = 2 and w3=w4 = 0.5

A bounded search is considered for the controller parameter values (ie Each parameter is bounded between a minimum and a maximum value) The heuristic search explores the five dimensional search space in order to identify the optimal solutions

4 Results and Discussions

In the proposed work, the initial algorithm parameters are assigned as follows: number of agents is chosen as

twenty, dimension of search is five, cost function is chosen as J min, maximum iteration number is chosen as 500 For each algorithm, the heuristic search is repeated ten times and the average value is chosen as the optimized value All the simulation work is carried using Matlab software The proposed work is tested on the following three FOPTD models In this work, for all the considered process models, a disturbance signal of 0.5 (50% of setpoint value) is considered to analyze the regulatory response of the designed 2DOF PID

Process 1: The first-order stable process model is considered [12,13]:

s 5 0 e 1 s

1 )

s

(

Heuristic algorithm based based controller design is initially proposed for the above process using the feedback

2DOF structure with the following bounded values 0% < K p < 50%, 0% < K i < 20%, 0% < K d < 50%, 0% < α < 100% and 0% < β < 100% During the search, the heuristic algorithm explores the search space based on the

controller range Ten independent runs are performed with each heuristic algorithm and the average value of the search is considered as optimal value In this process, the controller values and the corresponding performance

Heuristic algorithm Initial parameters

CF

+

Fig 3 Block diagram of proposed controller design procedure

values are presented in Table 1 and the corresponding response is presented in Fig 4 For this process, the BFO tuned 2DOF PID offers better response compared with other methods Similar response is obtained with the feed forward 2DOF structure

0 0.2 0.4 0.6 0.8 1

Time (sec)

PSO BFO CS FA

Fig 4 Reference tracking and disturbance rejection performance for Process 1 Table 1 Optimized controller parameters and the performance measure values

Process 2: The FOPTD model of the spherical tank system is given below [10,11]:

s 7 11 e 1 s 46 330 6215 3 )

s

(

The heuristic algorithm based search is proposed for this system, as discussed in process 1 The optimal controller parameters and the corresponding performance measurre values are presented in Table 1 Fig 5 depicts the servo and regulatory operation with a disturbance signal of 0.5 The CS based method offers satisfactory response compared with the alternatives

0 0.5 1 1.5

Time (sec)

Reference PSO BFO CS FA

Fig 5 Reference tracking and disturbance rejection for spherical tank system

Process 3: The first-order unstable process with the following transfer function model is considered [3,7]:

s 1 0 e 1 s 89 5

8644 5 )

s

(





Designing a suitable controller for unstable system is quite defficult compared with stable system In this work, the

controller bouundaries are assigned as -60% < K p < 0%, -25% < K i < 0%, -25% < K d < 0%, 0% < α < 100% and 0% < β < 100% The optimal controller parameters and the corresponding iteration number, M , and t are presented

in Table 1 From Fig 6, it is noted that, the FA tuned 2DOFPID offers better result compared with PSO, BFO and

CS,

0 0.5 1 1.5

Time (sec)

Reference PSO BFO CS FA

Fig 6 Reference tracking and disturbance rejection for unstable bioreactor model

5 Conclusions

In this paper, design of 2DOF PID controller design is proposed using PSO, BFO, CS and FA The proposed method

is tested on two stable FOPTD models and one unstable FOPTD model The proposed controller design procedure is

validated using traditional measures, such as M p and t s The results show that, number of iteration taken by the LF driven CS and FA is comparatively smaller than PSO and BFO The simulation result also confirms that, even though there is a structural difference, the feedback and feed forward 2DOF PID offers similar process response for

the servo and regulatory operations

References

1 Aidan O’Dwyer Handbook of PI and PID controller tuning rules, 3rd Edition, Imperial College Press, London, 2009

2 Araki M, Taguchi, H Two-Degree-of-Freedom PID controllers, International Journal of Control, Automation, and Systems, 2003, 1(4):

401-411

3 Bequette WB Process Control – Modeling, Design and Simulation, Prentice – Hall of India Pvt Ltd, 2003

4 Fister IJ, Yang XS, Fister I., Brest J, Fister D A brief review of nature-inspired algorithms for optimization, Electrotechnical Review, 2013,

80(3)

5 Kennedy J, Eberhart RC Particle swarm optimization In Proceedings of IEEE international conference on neural networks, 1995:

1942-1948

6 Latha K, Rajinikanth V 2DOF PID controller tuning for unstable systems using bacterial foraging algorithm Lecture Notes in Computer Science, 2012 LNCS (7677): 519–527

7 Padmasree R, Chidambaram M Control of Unstable Systems, Narosa Publishing House, India, 2006

8 Passino KM Biomimicry of bacterial foraging for distributed optimization and control, IEEE Control Systems Magazine, 2002, 22(3): 52-67

9 Rajinikanth V, Latha K Setpoint weighted PID controller tuning for unstable system using heuristic algorithm, Archives of Control Sciences,

2013, 22(4): 481–505

10 Rajinikanth V, Latha K Controller Parameter Optimization for Nonlinear Systems Using Enhanced Bacteria Foraging Algorithm, Applied Computational Intelligence and Soft Computing, Volume 2012, Article ID 214264, 12 pages, 2012

11 Sundaravadivu K, Arun B, Saravanan K Design of Fractional Order PID controller for liquid level control of spherical tank, IEEE

DOI: 10.1109/ICCSCE.2011.6190539

12 Vijayan V, Panda RC Design of PID controllers in double feedback loops for SISO systems with set-point filters, ISA Transactions, 2012,

51 (4): 514–521

13 Vijayan V, Panda RC Design of a simple setpoint filter for minimizing overshoot for low order processes, ISA Transactions, 2012, 51 (2): 271–276

14 Yang XS, Deb S Cuckoo search via Lévy flights, In: Proceeings of World Congress on Nature & Biologically Inspired Computing (NaBIC 2009), IEEE Publications, USA, 2009: 210-214

15 Yang XS Firefly algorithm, stochastic test functions and design optimisation, I Journal of Bio-inspired Computation, 2010, 2(2): 78-84

16 Yang XS Nature-Inspired Metaheuristic Algorithms, Luniver Press, UK, 2008

17 Raja NSM, Rajinikanth V, Latha K Otsu Based Optimal Multilevel Image Thresholding Using Firefly Algorithm, Modelling and Simulation

in Engineering, vol 2014, Article ID 794574, 17 pages, 2014